There are two results that "classify" cyclic Galois extensions of fields.
First one is Lagrange Theorem that holds under certain assumptions:
Let $K$ be a field $n \in \mathbb{N} \setminus \{0\}$.
For $p = Car(K) \neq 0$ (Car = characteristic) we may assume that $p \nmid n$.
We assume that there exists $\xi$ a nth root of unity
Then one has Artin-Schreier theorem that holds under assumptions:
Let $K$ be a field with $Car(K) = p$ prime
Is it not limiting to assume that there exists $\xi$ a primivite nth root of unity in the base field for the first case?